語言科目

這段改編自 2010 年 8 月 11 日的對話。

(TK: 通常,最勁(厲害)的人,都是讀數學出身的。彷彿只要數學勁,就幾乎什麼行業,都可以做得到。)

無錯。你知不知道,為什麼數學和中、英文一樣,都歸入「主科」?

漫畫化地講,因為三科也是「語言科目」。

中文,是中國人的語言;

英文,是地球人的語言;

而數學,則是宇宙人的語言。

「宇宙人語言」的意思是,在宇宙中的任何一個角落,一加一都是等於二。無論你移居去哪一個星球,你也不用擔心,你已學的數學知識,不再適用。數學可以說是,應用範圍取廣的知識。

中、英、數三科也是「語言科目」的意思是,它們不只是一般的知識,而且還直接是,其他科目的工具。其他知識中的大部分,也要用中、英、數來表達。

微觀而言,中、英、數的好壞,會直接影響到其他科的成績高低。宏觀來說,缺少其中一項語言的話,你將來的主修,或者專業的選項,就會大大減少。

而更嚴重的是,你的數學成績,會直接反映了,你智力的高低。假設你的數學一流,你很難可以說服到人,其實你的智力奇低。

— Me@2014.10.29

2014.10.29 Wednesday (c) All rights reserved by ACHK

Functional Paradox

paradox ~ mixing a level with its meta-level

There is no mixing-level problem for the equation

f(x) = x,

because it just means that two variables have the same value. In other words, the value of a function of x is equal to the value of x.

However, for the equation

f = x,

there is a mixing-level problem, resulting no meanings; because

the equation means that a function (aka a formula of a number) is equal to a variable (aka a number).

A formula of a number is a structure. It cannot be a number.

— Me@2013-07-16 10:45 AM

2014.10.13 Monday (c) All rights reserved by ACHK

Amazing Gags 8

這段改編自 2010 年 4 月 10 日的對話。

之前講過,「搞 gag 人士」有如「魔術師」,要遵守「魔法守則」,不可公開魔法要訣。魔術精采的原因,在於神秘感。你知道方法後,就不再精采。

但是,現在我發現,公開「搞 gag」(弄笑話)的方法,其實也沒有所謂。一來,有一大堆。二來,對一般人而言,即是知道方法,也很難會學得到。例如,其中一個方法是:

當你涉獵了大量的科目,再加上鑽研了其中的一部分,你就會有近乎,用之不盡的「搞 gag」靈感。

— Me@2014.10.11

2014.10.11 Saturday (c) All rights reserved by ACHK

Cartesian Dualism

In philosophy of mind, dualism is the position that mental phenomena are, in some respects, non-physical, or that the mind and body are not identical. Thus, it encompasses a set of views about the relationship between mind and matter, and is contrasted with other positions, such as physicalism, in the mind–body problem.

— Wikipedia on Dualism (philosophy of mind)

The mind–body problem in philosophy examines the relationship between mind and matter, and in particular the relationship between consciousness and the brain.

The problem was famously addressed by Rene Descartes in the 17th century, resulting in Cartesian dualism, and by pre-Aristotelian philosophers, in Avicennian philosophy, and in earlier Asian traditions. A variety of approaches have been proposed. Most are either dualist or monist. Dualism maintains a rigid distinction between the realms of mind and matter. Monism maintains that there is only one unifying reality, substance or essence in terms of which everything can be explained.

— Wikipedia on Mind–body problem

2014.10.10 Friday ACHK

背誦量

全像記憶 3

這段改編自 2010 年 8 月 11 日的對話。

(TK: 運算機會率題目時,如何提升準確度?) 

九成九是靠背誦 —— 背誦眾多運算方法,和萬千驗算技巧。當然,我不是要你「死背」,而是要你「生背」,即是明白以後才背。

千萬不要企圖,自己發明任何方法。一來,你未有那些智力。二來,即使有,你也負擔不到那些時間。

只有數學家才會,負擔得起那些智力,和那些時間。

(TK: 其實我是有背的,但是,時常也誤中副車,差一點才能想中正確方法。) 

或者說,你背得不夠多,或者不夠詳細。我所指的「背」,其實份量是十分驚人的。

例如,假設考試有可能出現的機會率題目,總共有 5 類。我並不是說,你每類也背誦一題的方法,就可以奪得好成績。

實際上,你的背誦量,並不只是 5 題,而隨時可能是 50 題,因為,同一種題目,可以有(例如)10 種不同的問法。

那 10 種題形的應對方法(和對應的驗算技巧),你都要背誦,因為,同一種題目,你要背誦了它,很多不同的版本,才會領略到,背後的精髓。那你才可以做到「明白以後才背」,即是「生背」。

如果你一定要成績奪 A,背誦量是十分驚人的。所以,我多次提醒你,你在每次做 past paper(以往試題),或其他練習之前,也一次要先背誦你的「魔法筆記」。

「背」的意思並不是說,你把「魔法筆記」,由頭至尾,閱讀一次就算。「背」的真正意思是,要你做到「過目不忘」,即是,在平日做練習,或者考試時,你都可以在心裡翻查,筆記上的每一頁,每一個細節。

— Me@2014.10.05

2014.10.06 Monday (c) All rights reserved by ACHK

Heisenberg

My first query is why does he claim the position and period of an electron to be unobservable “in principle”? There was theoretically no reason (at THAT time) to doubt that these quantities could be measured, though certainly they were indeterminate practically.

Werner Heisenberg obviously disagreed with this assumption of yours and it just happened that his ability to disagree made him a founder of quantum mechanics.

He has spent several years by trying to develop “quantized planetary” models of the helium atom etc. before he understood that this failing project is failing for fundamental reasons. Such a helium with well-defined positions would be described by a chaotic 3-body problem and there would be no way how it could be consistent with the known regular behavior of the helium atom (and other atoms and other coherent systems), including the sharp spectral lines.

So Heisenberg was able to see in 1925 something that you can’t see now: that the electrons can’t be going along any particular trajectories while they’re in the atoms. Instead, what is observed is that they have a totally sharp energy from a possible list, the spectrum – something we can really observe via the photons that atoms emit or absorb. To conclude that electrons can’t be going along particular classical trajectories in the atoms, he didn’t have to wait for measuring apparatuses that would be sufficiently accurate. He was able to make this conclusion out of the available data by “pure thought”, and he was right.

— Lubos Motl

2014.10.04 Saturday ACHK

天人天書 1.2.1

這段改編自 2010 年 4 月 10 日的對話。

(安:你彷彿收集了很多「神作」。你最初是怎樣知道,那些「神作」的存在?)

其實沒有什麼特別的原因。就正如我問,你是怎樣認識到,你現在的朋友?

那並沒有任何神奇的方法。你會「自然」遇到。

(安:但是,可以叫做「神作」,即是十分難得。換句話說,遇到的機會率較微。間中給你遇到一兩本「神作」,並不出奇。但是,你好像遇見過,大量的「神作」。那就十分奇怪。)

每一門學問中,第一本「神作」,是很難會遇到。但是,當你博覽群書後,總會遇到一兩本「神作」。而那一兩本「神作」,就自然會引薦其他「神作」給你。

— Me@2014.10.02

2014.10.02 Thursday (c) All rights reserved by ACHK

String theory rivals

Does string theory have rivals? The answer by the most cited theoretical physicist (and perhaps scientist) is that there are not any interesting competing suggestions. Be sure that people attending my popular talks (and sometimes radio hosts etc.) often ask the same question and I give the same answer. One reason, as Witten reminds us, is that ideas that actually have something good about them, like twistors and noncommutative geometry, are gradually identified as aspects of string theory itself and absorbed by string theory. It’s just how the things are.

— An interview with Edward Witten at a bizarre place

— Lubos Motl

2014.10.01 Wednesday ACHK